#428914

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,9,8)(7,11,12,10)(13,14,19,24)(15,25,29,36)(16,30,34,39)(17,23,18,22)(20,28,37,27)(21,33,40,32)(26,35,42,44)(31,43,41,38);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,27,19,29,18,28)(14,20,22,25,24,37,23,36)(16,31,32,26,34,41,33,42)(21,38,30,35,40,43,39,44);(2,4)(13,16)(14,21)(15,26)(17,33)(18,32)(19,34)(20,35)(22,39)(23,30)(24,40)(25,38)(27,31)(28,41)(29,42)(36,43)(37,44);(5,8,9,6)(7,10,12,11)(15,27)(17,18)(20,25)(22,23)(26,31)(28,29)(30,39)(32,33)(35,38)(36,37)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336110);